Vectors
What you must understand
You need fluency with vectors in two and three dimensions: addition, scalar multiplication, position vectors, displacement, magnitude, unit vectors, the dot product and its geometric meaning, and the cross product including its magnitude as an area and its direction by the right-hand rule. You must move confidently between component form and geometric reasoning, and use vectors to prove geometric facts (parallelism, collinearity, perpendicularity, concurrency) and to find lines and planes, distances and angles.
Traps
- Dot versus cross: (\mathbf{a}\cdot\mathbf{b}) is a scalar; (\mathbf{a}\times\mathbf{b}) is a vector. Mixing them up destroys a solution.
- Cross product order: (\mathbf{a}\times\mathbf{b} = -\mathbf{b}\times\mathbf{a}); reversing order flips the normal direction.
- Projection formula: the scalar projection of (\mathbf{a}) on (\mathbf{b}) is (\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|}); forgetting to divide by (|\mathbf{b}|) is a classic error.
- Collinearity: three points are collinear only if one displacement is a scalar multiple of another and they share a point.
- Angle range: the angle between vectors from the dot product lies in ([0,\pi]); a negative dot product signals an obtuse angle.
Rule of thumb
For geometry proofs, express every relevant point as a position vector from a single origin, then translate the geometric claim into an algebraic one (parallel = scalar multiple; perpendicular = zero dot product).
Worked example
Find the shortest distance from point (P) to the line through (A) with direction (\mathbf{d}). Let (\mathbf{v} = \overrightarrow{AP}). The perpendicular distance is [ \frac{|\mathbf{v}\times\mathbf{d}|}{|\mathbf{d}|}. ] In two dimensions this reduces to (\frac{|v_x d_y - v_y d_x|}{|\mathbf{d}|}).